integration of a strapdown gravimeter system in an ... -...

Integration of a strapdown gravimeter system in anAutonomous Underwater Vehicle

Clément ROUSSEL

PhD - Student (L2G - Le Mans - FRANCE)

April 17, 2015

Clément ROUSSEL ISPRS / CIPA Workshop April 17, 2015 1 / 24

Plan

1 Introduction

DefinitionInterests & ApplicationsPrinciple

2 Design & Equation

Instrumentation & CarrierEquation of moving-base gravimetry

3 Performance assessment & Filtering

Numerical simulationsFiltering strategy

4 Further work


Introduction

What is gravimetry ?

scalar gravimetry: ‖g‖

vectorial gravimetry:

gxgygz


Introduction

Why do we measure gravity in the subsea domain ?

in geodesy: to improve the determination of the geoid

in geophysics: to determine the distribution of masses in the ocean crust

in navigation: to improve underwater navigation


Introduction

How do we measure gravity in the subsea domain ?

Gravity is usually measured in units of acceleration [1m.s−2 = 10−5 mGal ]

An instrument used to measure gravity is known as a gravimeter

One can regard gravimeters as special-purpose accelerometers

Unmanned Underwater Vehicle (UUV)


Design & Equation

Instrumentation

LiMo-g : Light Moving gravimetry system

Geodesy and Geomatic Lab (L2G) & Geodesy Lab (LAREG)

Doctoral Thesis of Bertrand de Saint-Jean (2008)

GRAVIMOB : MOBile GRAVImetry system

Oceanic Domains Lab (LDO)

Marcia Maïa (Head scientist) & Jean-François d’Eu (Instrumentationengineer)


Design & Equation

Instrumentation

Strapdown sensor

Six electrostaticaccelerometers

Two triads (α & β) installedin a waterproof sphere ofabout 40 cm diameter


Design & Equation

Carrier

Autonomous Underwater Vehicule:AsterX

IFREMER: French ResearchInstitute for Exploitation of the Sea

Navigation: INS + DVL + USBL

able to dive down to 3, 000m &travel up to 100 km

total mass of 800 kg &scientific payload of 200 kg


Design & Equation

Equation of moving-base gravimetry

Application of Newton’s Second Law:

X iα = g i

α + aiα (1)

X iα position vector of the proof mass

Mα

X iα second-order derivative of X i

α

g iα gravitationnal acceleration

aiα restoring force per unit of mass

projected onto the i-frame


Design & Equation

Frames

i-frame inertial-frame

e-frame earth-frame (in rotation with respect to the i-frame)

n-frame navigation-frame (defined by the geographic coordinates of theAUV: longitude λP , latitude ϕP and ellipsoidal height hP)

b-frame body-frame (defined by the attitude angles of the AUV: heading δ,pitch χ and roll η)


Design & Equation

Equation of moving-base gravimetry

Only P position (λP , ϕP , hP) is known:

X iα = X i

P + Liα = C i

eXeP + C i

bLbα (2)

XP position vector of the vehiclereference point P

Lα lever arm PMα

C ie rotation matrix transforming

e-frame into i-frame

C ib rotation matrix transforming

b-frame into i-frame


Design & Equation

Equation of moving-base gravimetry (neglecting the rotation of the Earth)

gnα = Cn

e X eP + Cn

b

(Ωb

ebΩbeb + Ωb

eb

)Lbα − Cn

b abα (3)

gnα gravitational vector at point Mα

Cne rotation matrix transforming e-frame into n-frame (λP , ϕP)

X eP second-order derivative of X e

P (λP , ϕP)

Cnb rotation matrix transforming b-frame into n-frame (δP , χP , ηP)

Ωbeb skew symmetric matrix associated with the rotation of the b-frame with

respect to the e-frame (λP , ϕP , δP , χP , ηP)

Lbα lever arm PMb

α

abα restoring force per unit of mass


Design & Equation

Equation of moving-base gravimetry (neglecting the rotation of the Earth)

gnα = Cn

e X eP + Cn

b

(Ωb

ebΩbeb + Ωb

eb

)Lbα − Cn

b abα (4)

gnβ = Cn

e X eP + Cn

b

(Ωb

ebΩbeb + Ωb

eb

)Lbβ − Cn

b abβ (5)

And gnP ? Under the assumption that Lα = −Lβ :

gnP ≈

gnα + gn

β

2(6)

gnP ≈ Cn

e X eP − Cn

b

(abα + ab

β

2

)(7)


Performance assessment & Filtering

Numerical simulations (Monte-Carlo)

gnP ≈ Cn

e X eP − Cn

b

(abα + ab

β

2

)(8)

f :

λP , ϕP , hPδ, χ, ηaα, aβ

→ gP (9)

f multivariate function mapping R12 into R3




f :

λP + ελ, ϕP + εϕ, hP + εhδ + εδ, χ+ εχ, η + εη

aα + εα, aβ + εβ

→ gP + εg (10)

εθ additive noise term, θ = λ, ϕ, h, δ, χ, η

εg error on gravity vector

N random draws

E [εg ] =1N

N∑i=1

εg ,i (11)

σ[εg ] =√

E [(εg − E [εg ])2] (12)




Reference gravity field derives from a geological model of oceanic crust

bathymetric survey (2.70 g.cm−3)distribution of mineral blocks (3.85 g.cm−3)

Reference trajectory of the AUV

derives from a test mission carried out by the IFREMERdeterministic modelspolynomial & periodic functions12 profiles, each about 3, 600m long, hP = −2200m




3



Numerical simulations (Monte-Carlo): effect of the AUV positionning uncertainty

uncertainty = 0.1 % of thetravelled distance(manufacturer’sinformations)

ελ & εϕ are modelled by adouble integration of aGaussian White Noiseprocess

εh is modelled by a simpleGaussian White Noiseprocess

Results are consistent with the fact that the uncertainty affecting thecoordinates hP is more likely to perturb the restitution of the verticalcomponent gu



Numerical simulations (Monte-Carlo): effect of the AUV attitude uncertainty

uncertainty = 0.02deg (δ)& 0.01deg (χ & η)(manufacturer’sinformations)

εδ, εχ & εη are modelled bya Gaussian White Noiseprocess

Components g e & gn are more affected by the attitude uncertaintythan the vertical component gu



Numerical simulations (Monte-Carlo): effect of the accelerometer uncertainties

Stochastic processes areidentified thanks to AllanVariance

εα & εβ are modelled by aWhite Noise and a first orderRandom Walk processes

The low pass filtering has no effect on the noise reducing because ofthe non-stationnary nature of the first order random walk process.



Filtering strategy

Kalman Filter

only for linear system

Extented Kalman Filter

needs Jacobian matrix to be estimated (implementation errors &time-consuming)may lead to an improper estimation of the covariance matrix if thelinearity hypothesis is not respected (divergence of the filter)

Unscented Kalman Filter

does not require the calculation of Jacobian matrixrelies on a deterministic sampling method



Filtering strategy


Further Work

Calibration

scale factor & bias of each accelerometertransformation matrix Cb

s

Improve Unscented Kalman Filtering

complex noise modelsspatial variability of the gravity field

Test mission

scheduled on March 2016off the Mediterrean coasts in the south of France


Thank you for your attention

We are indebted to the French Ministry ofDefence and the Pays de la Loire Region fortheir support of this work.

Jérôme Verdun - supervisorMarcia Maïa - co-supervisorJosé CaliJean François d’Eu

[email protected]


integration of a strapdown gravimeter system in an ... -...

Documents